Explanation Given: The microbe concentration of Lake Tom depends on the oxygen concentration and it is expressed as, B ( x ) = x 3 − 7 x 2 − 160 x + 1800 , 0 ≤ x ≤ 20 Here, B ( x ) is the microbe concentration and x is the oxygen concentration. Formula used: Write the formula for derivative of power rule. If f ( x ) = x n for any number n , then the derivative. f ′ ( x ) = n x n − 1 If f ( x ) = k for any real number k , then the derivative is defined by the formula. f ′ ( x ) = 0 To find local maximum and minimum find the values of x from the equation f ′ ( x ) = 0 . Then find f ″ ( x ) and put the values of x. If f ″ ( x ) equals some positive quantity then it will be Local minimum is located at that point x and if f ″ ( x ) equals some negative quantity then Local maximum is located at the point at that point. Calculation: For the extremum microbe concentration, the derivative of the microbe concentration would be zero. Recall the expression of microbe concentration. B ( x ) = x 3 − 7 x 2 − 160 x + 1800 …… (1) Differentiate the expression of microbe concentration with respect to x by using the power rule for derivative. B ′ ( x ) = 3 x 3 − 1 − 7 ( 2 x 2 − 1 ) − 160 ( 1 ) + 0 = 3 x 2 − 14 x − 160 …… (2) For the extremum microbe concentration, B ′ ( x ) = 0 . Substitute 0 for B ′ ( x ) . 3 x 2 − 14 x − 160 = 0 Simplify the equation. 3 x 2 − 30 x + 16 x − 160 = 0 3 x ( x − 10 ) + 16 ( x − 10 ) = 0 ( x − 10 ) ( 3 x + 16 ) = 0 Solve for x

MyLab Math with Pearson eText -- Standalone Access Card -- for Mathematics with Applications in the Management, Natural, and Social Sciences (12th Edition)

12th Edition

ISBN: 9780134856551

Author: Margaret L. Lial, Thomas W. Hungerford, John P. Holcomb, Bernadette Mullins

Publisher: PEARSON

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Chapter 12.1, Problem 55E

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To calculate: The microbe concentration of Lake Tom depends on the oxygen concentration and it is expressed as B(x)=x3−7x2−160x+1800, 0≤x≤20. Here, B(x) is the microbe concentration and x is the oxygen concentration.

The oxygen concentration that will lead to the minimum microbe concentration.

The minimum microbe concentration.

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Chapter 12.1, Problem 53E

Chapter 12.1, Problem 59E

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MyLab Math with Pearson eText -- Standalone Access Card -- for Mathematics with Applications in the Management, Natural, and Social Sciences (12th Edition)

Ch. 12.1 - Checkpoint 1 For what values of x is the function...Ch. 12.1 - Checkpoint 2 Find all intervals on which is...Ch. 12.1 - Checkpoint 3 Identity the x-values of all points...Ch. 12.1 - Checkpoint 4 Find the critical numbers for each of...Ch. 12.1 - Prob. 5CP Ch. 12.1 - Prob. 6CP Ch. 12.1 - Checkpoint 7 Find the locations of the local...Ch. 12.1 - Prob. 8CP Ch. 12.1 - Checkpoint 9 If a sales function is given by...Ch. 12.1 - Prob. 1E

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Solution Summary: The author explains how the microbe concentration of Lake Tom depends on the oxygen concentration.

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